Know the enemy: 2D Fermi liquids Sankar Das Sarmaa, Yunxiang Liaoa aCondensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, MD 20742, USA Abstract We describe an analytical theory investigating the regime of validity of the Fermi liquid theory in inter- acting, via the long-range Coulomb coupling, two-dimensional Fermi systems comparing it with with the corresponding 3D systems. We find that the 2D Fermi liquid theory and 2D quasiparticles are robust up to high energies and temperatures of the order of Fermi energy above the Fermi surface, very similar to the cor- responding three-dimensional situation. We calculate the phase diagram in the frequency-temperature space separating the collisionless ballistic regime and the collision-dominated hydrodynamic regime for 2D and 3D interacting electron systems. We also provide the temperature corrections up to third order for the renor- malized effective mass, and comment on the validity of 2D Wiedemann-Franz law and 2D Kadawoki-Woods relation. Keywords: Fermi liquid, electron self-energy, random phase approximation Contents 1 Introduction 2 2 Theory 3 2.1 General formulas for electron self-energy . .3 2.1.1 Keldysh approach to interacting electrons . .3 2.1.2 RPA dynamically screened interaction . .5 2.1.3 RPA self-energy . .7 2.2 2D electron self-energy . 10 2.2.1 Momentum integration . 10 2.2.2 Frequency integration . 13 2.3 3D electron self-energy . 15 2.4 Discussion of the analytical results . 16 3 Results 17 3.1 Applicability of the FL theory . 17 3.2 Effective mass . 22 3.3 Hydrodynamic and ballistic regimes . 23 4 Wiedemann-Franz (WF) and Kadowaki-Woods (KW) Relations for 2D Interacting Sys- tems 25 arXiv:2101.07802v1 [cond-mat.str-el] 19 Jan 2021 4.1 WFLaw................................................ 26 4.2 KWlaw................................................ 27 5 Conclusion 27 A Derivation of self-energy formulas using Matsubara technique 28 Preprint submitted to Annals of Physics January 21, 2021 B Integrals involving hyperbolic functions 29 1. Introduction In a famous talk at the 1989 Kathmandu Summer School, Phil Anderson questioned the validity of the Fermi liquid theory and the applicability of the quasiparticle concept to high-temperature cuprate superconductors specifically and to 2D interacting electron systems generally [1]. One of his lectures, which was also quoted in his published lecture note, has the memorable slogan \Know the enemy", alluding specifically to the Fermi liquid theory as the enemy. Anderson was the first person vehemently and tirelessly pushing the idea that the 2D physics of cuprate superconductors is beyond the Fermi liquid-BCS paradigm, and represents new physics, which is now commonly referred to as non-Fermi liquids, a terminology virtually unknown in 1989. The basic challenges Anderson posed were simple: (1) Is it possible that interactions destroy the 2D Fermi surface just as they do in one dimension leading to a Luttinger liquid? (2) Is it possible that cuprates represent a new emergent form of superconductivity which simply cannot be explained and understood using the highly successful Fermi liquid-BCS formalism of electron pairing around the Fermi surface leading to a superconducting instability of Copper pairs condensing into a BCS ground state? Amazingly, there is still no answer to the second question even after 30 years and many thousands of theoretical papers as there is no consensus on the accepted theory of cuprate superconductivity. In fact, even the precise mechanism for cuprate superconductivity is still actively debated in the theoretical community, and Anderson himself worked on developing an appropriate theory for the cuprates for the rest of his life. It is, however, a great testimonial to Anderson's early insight that most theorists working on cuprate superconductivity accept that its explanation most likely lies outside the standard Fermi liquid-BCS paradigm. But the answer to Anderson's first question is definitively known. Two dimensional interacting Fermi systems are Fermi liquids similar to 3D interacting Fermi systems, and not non-Fermi liquids like interacting one dimensional Luttinger liquids. In fact, Anderson's trenchant questioning of the nature of interacting 2D electron systems (\Know the enemy") led to many theoretical developments establishing conclusively, and in fact, even rigorously, that interacting 2D electron systems are indeed normal Fermi liquids with well-defined renormalized Fermi surfaces very much like 3D normal metals and normal He-3 [2, 3, 4]. The basic idea, which is now established with reasonable mathematical rigor, is rather simple and essentially the same as it is for 3D interacting Fermi systems. The imaginary part of the interacting 2D self-energy at 2 energy E goes as (E − EF) up to some logarithmic corrections, with EF being the Fermi energy, and this holds to all orders in perturbation theory. Thus, the single-particle spectral function is a delta function at the Fermi surface, guaranteeing the existence of an interacting Fermi surface and low-energy quasiparticles with one to one correspondence to the noninteracting Fermi gas. This behavior is similar to 3D systems within logarithmic accuracy, and thus 2D systems are similar to 3D systems. This is very different from interacting 1D systems, which within the same perturbation theory gives an imaginary self-energy going 1=2 as (E − EF )(ln(E − EF )) [5]. Although a perturbation theory is neither meaningful nor valid for a 1D system, it is clear that the Fermi surface cannot exist in 1D already based on this simple perturbative argument. On the other hand, the perturbative analysis applies in 2D and shows that Fermi surface exists in the 2D interacting systems just as in 3D. In the current work, we extend Anderson's first question, trying to understand `the enemy' better. The question we ask is the extent to which the Fermi liquid theory applies in 2D electron liquids in the presence of long-range Coulomb interactions. In particular, how far in energy from the Fermi surface and how high in temperature can we go and still find well-defined quasiparticles in 2D interacting systems? What is the regime of applicability of the concept of 2D Fermi liquids? We answer these questions analytically, both in 2D and 3D comparing the two situations, using a many-body perturbation theory which is exact for the Coulomb-interacting system in the high-density limit. We also calculate the temperature dependent effective mass renormalization, comment on the 2D Wiedemann-Franz law and Kadowaki-Woods relation, and estimate the hydrodynamic regime in Fermi systems interacting through long-range Coulomb interactions. 2 Our work is completely analytical involving expansions in inverse density, energy, and temperature on an equal footing. The theory itself uses the well-established leading-order dynamical screening or random phase approximation (RPA) for the self-energy, which is exact in the high-density limit. 2. Theory In this section, we derive the self-energy for an electron system with long-range Coulomb interactions in both 2D and 3D [6, 7, 8, 9]. In particular, we provide the analytical expressions for both the real and imaginary parts of the on-shell self-energy, for arbitrary energy-to-temperature ratio "=T . Here, "(= E −EF) is the quasiparticle energy measured from the Fermi surface and T is the temperature. The result is valid to the leading order in rs and to several orders in "=EF and T=TF. Here, rs is the standard dimensionless Coulomb coupling parameter (the ratio of the interparticle separation to the effective Bohr radius), and TF is the noninteracting Fermi temperature. As usual, rs is the many body perturbation parameter (or the density-dependent effective fine structure constant) for the theory which is strictly valid only for rs 1, but in practice works well empirically for metallic electron densities (rs ∼ 3 − 6). In Sec. 2.1, we start with reviewing the derivation of general self-energy formulas which are expressed as integrals involving RPA dynamically screened interaction and are used to extract the explicit self-energy expressions. Sec. 2.2 is devoted to the detailed calculation of the 2D electron self-energy, which is presented before in Ref. [9] and reviewed here for completeness. In Sec. 2.3, we provide the analytical expressions for the 3D electron self- energy with arbitrary "=T , which, to the best of our knowledge, are new for long-range Coulomb interactions. See Refs. [10, 11, 12] for the results of self-energy for the case of short-range interactions in both 2D and 3D. We also present the subleading terms in "=EF (T=TF) for the 3D self-energy in the " T (T ") limit. 2.1. General formulas for electron self-energy 2.1.1. Keldysh approach to interacting electrons In the framework of Keldysh formalism (see Ref. [13] for a review), we now rederive the general formulas for self-energy which are then used to obtain the analytic expressions presented in Secs. 2.2 and 2.3. An alternative derivation employing Matsubara technique (see for example Refs. [6, 7]) will be reviewed in Appendix A. We start with the partition function Z of a d−dimensional electron system with Coulomb interactions on a Keldysh contour which runs from t = −∞ to t = 1 and back to t = −∞. The system is assumed to be in the thermal equilibrium and noninteracting at the distant past t = −∞, after which the interactions are then adiabatically switched on. In terms of coherent state functional integral, the partition function can be expressed as Z ¯ Z = D ; exp (iS0 + iSint) ; (2.1) where S0 and Sint, which stand for the free and interacting parts of the action, respectively, are given by Z 1 Z 1 Z Z 0 d d 0 ¯ ^−1 0 0 0 0 S0 = dt dt d r d r (r; t) G0 (r − r ; t − t ) (r ; t ); −∞ −∞ Z 1 Z (2.2) 1 X d a a 0 0 a 0 a S = − ζ dt d r ¯ (r; t) ¯ 0 (r ; t)V (r − r ) 0 (r ; t) (r; t): int 2 a σ σ σ σ a=± −∞ Here +( ¯+) and −( ¯−) represent Grassmann fields residue on the forward and backward paths of the Keldysh contour, respectively.